Problem: Solve for $x$ : $ 6|x + 7| - 10 = -5|x + 7| + 7 $
Answer: Add $ {5|x + 7|} $ to both sides: $ \begin{eqnarray} 6|x + 7| - 10 &=& -5|x + 7| + 7 \\ \\ { + 5|x + 7|} && { + 5|x + 7|} \\ \\ 11|x + 7| - 10 &=& 7 \end{eqnarray} $ Add ${10}$ to both sides: $ \begin{eqnarray} 11|x + 7| - 10 &=& 7 \\ \\ { + 10} &=& { + 10} \\ \\ 11|x + 7| &=& 17 \end{eqnarray} $ Divide both sides by ${11}$ $ \dfrac{11|x + 7|} {{11}} = \dfrac{17} {{11}} $ Simplify: $ |x + 7| = \dfrac{17}{11}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 7 = -\dfrac{17}{11} $ or $ x + 7 = \dfrac{17}{11} $ Solve for the solution where $x + 7$ is negative: $ x + 7 = -\dfrac{17}{11} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& -\dfrac{17}{11} \\ \\ {- 7} && {- 7} \\ \\ x &=& -\dfrac{17}{11} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $11$ $ x = - \dfrac{17}{11} {- \dfrac{77}{11}} $ $ x = -\dfrac{94}{11} $ Then calculate the solution where $x + 7$ is positive: $ x + 7 = \dfrac{17}{11} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& \dfrac{17}{11} \\ \\ {- 7} && {- 7} \\ \\ x &=& \dfrac{17}{11} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $11$ $ x = \dfrac{17}{11} {- \dfrac{77}{11}} $ $ x = -\dfrac{60}{11} $ Thus, the correct answer is $x = -\dfrac{94}{11} $ or $x = -\dfrac{60}{11} $.